Despite many efforts to find a reasonable explanation, the origin of the
"knee" in the cosmic ray spectrum at energy around 10**15.5 eV remains
mysterious. In this letter we suggest that the "knee" may be due to a GZK-like
effect of cosmic rays interacting with massive neutrinos in the galactic halo.
Simple kinematics connects the location of the "knee" with the mass of the
neutrinos, and, while the required interaction cross section is larger than
that predicted by the Standard Model, it can be accommodated by a small
neutrino magnetic dipole moment. The values for the neutrino parameters
obtained from the analysis of existing experimental data are compatible with
present laboratory bounds.Comment: 8 pages, 1 figure, uses RevTe

Dova, M. T.

Epele, L. N.

Swain, J. D.

English

arXiv

Despite many efforts to find a reasonable explanation, the origin of the ”knee” in the cosmic ray spectrum at E ≈ 10015.5 eV remains mysterious. In this letter we suggest that the ”knee” may be due to a GZK-like effect of cosmic rays interacting with massive neutrinos in the galactic halo. Simple kinematics connects the location of the ”knee” with the mass of the neutrinos, and, while the required interaction cross section is larger than that predicted by the Standard Model, it can be accommodated by a small neutrino magnetic dipole moment. The values for the neutrino parameters obtained from the analysis of existing experimental data are compatible with present laboratory bounds.Facultad de Ciencias Exacta

Dova, María Teresa

Epele, Luis Nicolás

Swain, J.D.

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Massive relic neutrinos in the galactic halo and the knee in the cosmic ray spectrum
M.T.Dova and L.N.Epele
Departamento de F´ısica, Universidad Nacional de La Plata, C.C.67 - 1900 La Plata, Argentina
J.D.Swain
Department of Physics, Northeastern University, Boston, MA 02115
Despite many efforts to find a reasonable explanation, the origin of the ”knee” in the cosmic
ray spectrum at E ≈ 1015.5eV remains mysterious. In this letter we suggest that the ”knee”
may be due to a GZK-like effect of cosmic rays interacting with massive neutrinos in the galactic
halo. Simple kinematics connects the location of the ”knee” with the mass of the neutrinos, and,
while the required interaction cross section is larger than that predicted by the Standard Model, it
can be accommodated by a small neutrino magnetic dipole moment. The values for the neutrino
parameters obtained from the analysis of existing experimental data are compatible with present
laboratory bounds.
The cosmic ray (CR) spectrum is well-described by a power law of the E−γ form with a spectral index , which is
nearly constant over rather wide ranges of energy E, but displays significant discontinuities. The most notable occurs
in the region of the so-called ”knee” [1–7], at about 1015.5 eV, where the spectral index changes from 2.75 to 3.
Several models have been proposed to explain this ”knee”, none of which has been particularly convincing. They
include a change in CR sources from Type I to Type II supernovae and/or to changes of the particle acceleration
efficiency as a function of the electric charge of the primary [8–10]. Other proposals associate changes of the spectral
index with a reduction in the efficiency of the galactic magnetic field to confine the CR’s [11,12], allowing those at
higher energy to leak away. None of these models is able to reproduce the sharpness of the ”knee”. Another, rather
controversial, explanation of this kink in the spectrum is to attribute it to a recent, strong single source [13].
In this Letter, we propose that the ”knee” may be due to inelastic collisions between primary cosmic ray protons
and the cosmic neutrino background. For light Standard Model neutrinos with mass (mν < 1MeV ), Big Bang
cosmology predicts a thermal neutrino background of temperature Tν = 1.6×10
−4eV with an average number density
of nν ≈ 337cm
−1 , but which can rise dramatically in galaxies due to gravitational clustering provided that the
neutrinos are sufficiently massive (non-relativistic).
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The idea is a simple extension of the GZK mechanism [14,15] whereby protons above about 1019eV rapidly lose
energy via inelastic collisions with cosmic background radiation photons, but now considering the cosmic neutrino
background instead. Even allowing for the possibility that neutrinos have mass and cluster significantly, it is easy
to show that the Standard Model weak interaction cross sections are too small for these neutrinos to explain any
structure as pronounced as the ”knee”. Fortunately, allowing for a nonzero neutrino mass also immediately opens up
the possibility that neutrinos might carry magnetic dipole moments which could significantly increase the inelastic
proton neutrino cross section. Effectively we propose a softer GZK-like effect where interaction with the cosmic
microwave background radiation is replaced by interaction with virtual photons coupling to the magnetic dipole
moment of massive neutrinos. Combined with the increase in relic neutrino density due to gravitational clustering,
even rather modest values of magnetic dipole moment turn out to fit the experimental data of the KASCADE
experiment [16] rather well.
The required neutrino mass is set by simple kinematics from the requirement that it be sufficiently large that the
process ν + p → ν + ∆, ∆ → p + pi, can take place. Including higher resonances is not expected to change the
results significantly [21].
In order to calculate the inelastic proton-neutrino cross section in the relevant kinematics regime, we use the
parameterization of the measured quasi-elastic (i.e. including, and dominated by the ∆) cross section from [17] and
adapt it to the case of a neutrino with a magnetic moment. This necessitates replacing the usual electromagnetic
coupling −ieγµ with the appropriate derivative coupling κ
2mν
σµνq
ν where κ is the magnetic moment. A reasonably
straightforward calculation described in more detail elsewhere [22] then gives the required cross section as a function
of both mν and κ. A (CP-violating) electric dipole moment would have the same effect, and would be modeled with
the same sort of term with an additional factor of γ5 .
Fixing the threshold energy for the pion production at Ep = 3 × 10
15eV, one finds mν ≈ 100 eV. Laboratory
experiments clearly rule out such a mass for the electron neutrino, although the muon and tau neutrinos are still
viable candidates, as would be some other hypothetical neutrino-like dark matter particle. While neutrino oscillation
experiments suggest the possibility of a lighter neutrino, and cosmological arguments also tend to favour lighter
neutrinos, this rather large mass is by no means excluded. Of related interest is the ”Z-burst model”, which seeks to
explain the origin of CR events with energies above the GZK cutoff by interaction of ultrahigh energy neutrinos with
massive neutrinos in the galactic halo, producing bosons which then decay and give rise to the particles which we
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observe on earth (see the recent review [18] and references therein). This model favours a much lower neutrino mass,
but both the Z-burst model and the explanation we propose here can peacefully coexist, assuming different neutrino
species (of different masses) are responsible for the two effects.
To estimate the effects of gravitational clustering, we use the well accepted galactic halo mass distribution model
from [19] where the halo is described by the spheroidal density distribution. This assumes a uniform distribution of
neutrinos in a core of 10 kpc, and for mν = 100 eV yields nν = 1.4 × 10
8cm−3, compatible with bounds due to the
Pauli exclusion principle and the Tremaine-Gunn phase-space density constraints [20].
Following earlier work [21] on energy loss of cosmic rays in the cosmic microwave background, we find an energy
loss rate of
dE/dt =
c
γ
Z wm
0
dwoK(wo)σ(wo)n (1)
where w stands for the neutrino final energy in the proton rest frame, n is neutrino density, σ the proton-neutrino
cross section calculated above, and K is the average energy loss of the nucleon in the collision.
This equation is solved numerically, with the neutrino mass, magnetic dipole moment and propagation time as
parameters. Conservation of the number of nucleons is enforced by a balance equation
∂N
∂t
=
∂[bN ]
∂E
+D ∇2N +Q (2)
Here b(E) is the mean rate at which particles lose energy. The diffusion effect due to galactic magnetic field, modeled
by the D ∇2N is approximated here from the galactic residence time for CR’s, calculated from a diffusion model with
a containment volume which extends out 10 kpc from the core. The range of values we consider here is limited from
above by the age of the galaxy (1010 years) and from below by 108 years from the expected residence time derived
from spallation [23]. The third term corresponds to the particle injection rate which is assumed to have a power law
behavior, so that Q = KE−γ . The solution of this equation can be obtained in the same manner as in reference [21].
Figure 1 shows the total cosmic ray differential flux as measured by KASCADE together with the modified total
energy spectrum which is obtained from a sum of a proton component (abundance ≈ 60%) plus an iron component,
both with spectral index γ = 2.8 and a curve for the fitted value of κ = (5.4 ± 0.6)× 10−6µB, where µB is the Bohr
magneton assuming a residence time of 3 × 108 years. Different values of κ and t fit equally well provided that κ2t
is held constant. We use a simple superposition model to estimate the energy losses suffered by iron nuclei due to
interactions with neutrinos in the halo so that any effect in the iron spectrum should also be present at higher energies
3
scaling with A .
Interestingly, the corresponding soft GZK-like cutoff for the heavy component is expected above 1017eV, consistent
with a possible ”second knee” [24]. The KASCADE collaboration has shown [25] that the ”knee” is dominated by
the light component (≈ 70 %) of the CR with an energy dependent mass composition favouring a decrease of light
elements above the ”knee”. They have also shown that the light and heavy mass groups have comparable slopes up
to the ”knee” region, but beyond this energy the light component follows a steeper curve.
The heavy mass composition shows no significant ”knee” with a constant index γ . The calculated spectrum
successfully reproduces the KASCADE data in the region of the sharp ”knee”, with abundances matching well with
those estimated by the collaboration. The differences between the predicted and observed fluxes over 1016eV may
be attributable to the fact that in a more rigorous calculation the leakage of cosmic rays from the galaxy cannot
be completely neglected, while we assume here, as a first approximation, that the residence time is independent of
energy. In any case, the sharpness of the ”knee” in the spectrum cannot be explained by smooth analytic effects of a
transition between a regime dominated by diffusive propagation in the galactic magnetic field and ones in which the
escape of cosmic rays from the galaxy are suppressed (galactic modulation models).
In summary, we have considered the effect of CR particles interacting with massive neutrinos in the halo as an
explanation of the ”knee” in the cosmic ray spectrum. We have developed a novel approach which allows us to obtain
information about the properties of relic neutrinos in the galactic halo, as well as about their masses and magnetic
dipole moments. We are able to reproduce the KASCADE data around the ”knee” with a mixed composition of
protons and iron nuclei, with a sharp cutoff in the light component which is compatible with experimental data.
Results of detailed fits will be presented elsewhere, together with discussions of the possibilities for earth-based
accelerator experiments to study such neutrinos (or perhaps other massive candidate particles). So far, the best fit
results are not far from the present direct limits [26] set at accelerators (5 × 10−7µB), making future prospects very
interesting indeed!
ACKNOWLEDGMENTS
We would like to thank our colleagues in the Pierre Auger Collaboration for many stimulating discussions, and in
particular Lucas Taylor with whom early aspects of this idea were discussed. This work was supported by CONICET
(Argentina) and the National Science Foundation (USA).
4
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6
Figure Caption
Figure 1: Differential cosmic ray spectrum as observed bt KASCADE. Solid line shows the result of the fit with a
mixed proton-iron component.
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Massive relic neutrinos in the galactic halo and the knee in the cosmic ray spectrum

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Massive relic neutrinos in the galactic halo and the knee in the cosmic ray spectrum

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